Energy Storage and Duality in Circuits

Every reactive circuit you'll ever analyze — snubbers, decoupling networks, filters, resonant tanks — reduces to two energy-storage elements, capacitors and inductors, whose behavior is described by a strikingly symmetric pair of equations. Recognizing that symmetry (duality) isn't academic decoration: it lets you reuse every capacitor result for inductors for free, and it explains why series RLC and parallel RLC circuits are mirror images of each other in the math even though they look nothing alike on a schematic.

Energy Storage: The Two Mechanisms

A resistor dissipates energy irreversibly as heat: P = I²R. Capacitors and inductors instead store energy in a field and can return it to the circuit lossless (ideal case).

Capacitor — electric field storage

i = C·dv/dt
E_C = ½·C·V²

Energy is stored in the electric field between the plates, proportional to charge and voltage. Q = C·V, so E_C = ½·Q²/C = ½·C·V². Voltage across a capacitor cannot change instantaneously without infinite current — capacitor voltage is a state variable, continuous in time.

Inductor — magnetic field storage

v = L·di/dt
E_L = ½·L·I²

Energy is stored in the magnetic field of the flux linkage. Φ = L·I, so E_L = ½·Φ²/L = ½·L·I². Current through an inductor cannot change instantaneously without infinite voltage — inductor current is the state variable, continuous in time.

Both elements are memoryless in power but not in energy: instantaneous power p = v·i can be positive (charging, absorbing energy) or negative (discharging, returning energy), but the stored energy itself is a continuous function of the state variable and can never jump.

The Duality Table

Duality means every equation for one element has a structurally identical counterpart for the other, with roles of voltage/current, series/parallel, and L/C swapped.

The mirroring is exact: wherever a capacitor formula has C and v, swap in L and i and you get the inductor formula. This is why series RLC circuits (current is common, voltage sums — KVL-based) and parallel RLC circuits (voltage is common, current sums — KCL-based) produce dual transfer functions, dual resonance behavior, and dual Q-factor definitions.

Why Series Capacitors Add as 1/C and Series Inductors Add as L

This trips people up until you look at the charge/flux bookkeeping.

Series capacitors: same current flows through both, so the same charge Q accumulates on each. Total voltage is the sum: V = Q/C₁ + Q/C₂ = Q(1/C₁ + 1/C₂), so 1/C_eq = 1/C₁ + 1/C₂ — exactly like series resistors adding directly, but inverted, because capacitance is charge-per-volt (a "conductance-like" quantity for charge storage).

Series inductors: same current flows through both, same di/dt, so voltages add directly: V = L₁di/dt + L₂di/dt = (L₁+L₂)di/dt → L_eq = L₁ + L₂, just like series resistors.

This is the duality principle in action: inductors in series behave like resistors in series; capacitors in series behave like resistors in parallel, and vice versa for the parallel case.

Worked Example: Energy Handoff in an LC Tank

Take an ideal LC circuit: L = 10 mH, C = 100 nF, initially charged to V₀ = 12 V on the capacitor, zero initial inductor current.

Total energy (conserved, no resistance):

E_total = ½·C·V₀² = ½ × 100×10⁻⁹ × 12² = ½ × 100×10⁻⁹ × 144 = 7.2 µJ

Resonant frequency:

ω₀ = 1/√(LC) = 1/√(10×10⁻³ × 100×10⁻⁹) = 1/√(1×10⁻⁹) = 1/(31.6×10⁻⁶) ≈ 31.6×10³ rad/s
f₀ = ω₀/2π ≈ 5.03 kHz

Peak inductor current (all energy transferred to L at quarter period, since energy is conserved):

½·L·I_peak² = 7.2 µJ
I_peak = √(2 × 7.2×10⁻⁶ / 10×10⁻³) = √(1.44×10⁻³) ≈ 37.9 mA

Check: at the instant all energy is in the inductor, capacitor voltage is zero and vice versa — energy sloshes back and forth every quarter cycle, T/4 = 1/(4f₀) ≈ 49.7 µs. Verify total energy at an arbitrary phase, say ωt = 45°, where v_C = V₀cos(ωt) and i_L = (V₀/√(L/C))·sin(ωt):

E_C = ½C(V₀cos45°)² = ½C·V₀²·0.5 = 3.6 µJ
E_L = ½L·I_peak²·sin²45° = 3.6 µJ
Sum = 7.2 µJ ✓ (matches E_total, confirming conservation)

This is the textbook picture of a lossless resonant tank: energy continuously trades between electric and magnetic domains at f₀, with the two energies exactly 90° out of phase — direct evidence of the C↔L duality, since v_C and i_L play mathematically identical roles shifted in phase.

Practical Implications

• Decoupling capacitors rely on i = C dv/dt: a capacitor's ability to supply a fast current transient without voltage droop scales with C and with how fast di/dt demands change — this is why bulk and ceramic caps are sized differently for low- vs. high-frequency transients.
• Inductor-based supplies (buck/boost converters) rely on v = L di/dt: the inductor current ramps linearly under a fixed voltage, storing energy during the on-phase and releasing it during the off-phase — direct use of E_L = ½LI².
• Snubbers and clamps exist because real inductors demand somewhere for their stored current to go when a switch opens abruptly (di/dt → ∞ otherwise) — a flyback diode or RC snubber provides the dual path.
• State variables matter for simulation: SPICE transient solvers track capacitor voltages and inductor currents as the state vector precisely because these are the quantities that must remain continuous; everything else (resistor voltages, currents) is solved algebraically at each timestep.
• Duality speeds up hand analysis: if you've derived a filter response for a series RC high-pass, the dual parallel RL circuit's response follows immediately by swapping R↔G and C↔L in the transfer function — no need to re-derive from scratch.

Key Takeaways

• Capacitors store energy in an electric field (E = ½Cv²); inductors store it in a magnetic field (E = ½Li²) — both are lossless in the ideal case, unlike resistors.
• Capacitor voltage and inductor current are the state variables of a circuit: they must be continuous in time; their derivatives (current for C, voltage for L) can jump.
• Duality maps v↔i, C↔L, series↔parallel — every capacitor law has a structurally identical inductor law with these substitutions.
• Series capacitors combine like parallel resistors (1/C_eq = Σ1/Cᵢ); series inductors combine like series resistors (L_eq = ΣLᵢ) — the inverse of what many people assume by analogy to R.
• In a lossless LC tank, energy oscillates between electric and magnetic storage at ω₀ = 1/√(LC), with v_C and i_L 90° out of phase — a direct, verifiable illustration of the duality principle.